Chen Dissertation

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UNIVERSITY OF CALIFORNIA

Los Angeles

Fault Detection Filters

for

Robust Analytical Redundancy

A dissertation submitted in partial satisfaction of the

requirements for the degree Doctor of Philosophy

in Mechanical Engineering

by

Robert Hsu Chen

2000


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cCopyright by

Robert Hsu Chen

2000


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The dissertation of Robert Hsu Chen is approved.

D. Lewis Mingori

James S. Gibson

Fernando Paginini

Randal K. Douglas

Jason L. Speyer, Committee Chair

University of California, Los Angeles

2000

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To my parents

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TABLE OF CONTENTS

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

1 Introduction 1

1.1 Fault Detection Filter Background . . . . . . . . . . . . . . . . . . . 3

1.1.1 Fault Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.1.2 Beard-Jones Detection Filter . . . . . . . . . . . . . . . . . . . 5

1.1.3 Restricted Diagonal Detection Filter . . . . . . . . . . . . . . 10

1.1.4 Unknown Input Observer . . . . . . . . . . . . . . . . . . . . . 12

1.1.5 Approximate Unknown Input Observer . . . . . . . . . . . . . 141.2 Overview of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . 16

2 A Generalized Least-Squares Fault Detection Filter 21

2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.3 Conditions for the Nonpositivity of the Cost Criterion . . . . . . . . . 29

2.4 Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 Properties of the Null Space ofS . . . . . . . . . . . . . . . . . . . . 38

2.6 Reduced-Order Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.7 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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3 Optimal Stochastic Fault Detection Filter 51

3.1 System Model and Assumptions . . . . . . . . . . . . . . . . . . . . . 52


3.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.5 Perturbation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5.1 Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Robust Multiple-Fault Detection Filter 84

4.1 System Model and Assumptions . . . . . . . . . . . . . . . . . . . . . 85


4.3 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.4 Limiting Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5 Minimization with respect to H . . . . . . . . . . . . . . . . . . . . . 99

4.6 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Conclusion 118

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

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LIST OF FIGURES

2.1 Frequency response from both faults to the residual . . . . . . . . . . 47

2.2 Frequency response from the target fault and sensor noise to the resi-

dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.3 Time response of the residual . . . . . . . . . . . . . . . . . . . . . . 49

3.1 Frequency response from both faults to the residual . . . . . . . . . . 71

3.2 Time response of the residual . . . . . . . . . . . . . . . . . . . . . . 72

3.3 Eigenvalues of the Riccati matrix P and the filter for different Q1 . . 73

3.4 Frequency response from the target fault to the residual . . . . . . . . 73

4.1 Frequency response of the three single-fault filters . . . . . . . . . . . 103

4.2 Frequency response of the multiple-fault filter when s = 3 . . . . . . . 104

4.3 Frequency response of the multiple-fault filter when s = 2 . . . . . . . 105

4.4 Frequency response of the multiple-fault filter when s = 2 for different

Q1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.5 Frequency response of the two single-fault filters and the multiple-fault

filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.6 Frequency response of the multiple-fault filter under plant uncertain-

ties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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ACKNOWLEDGMENTS

I would like to thank Professor Jason L. Speyer for his patience, guidance and

encouragement through the years I have been at UCLA. I have benefited immensely

by being his student. I would like to thank Dr. Randal K. Douglas for sharing his

insight and expertise especially in the field of fault detection and identification and

for his continued support. I would also like to thank Professor D. Lewis Mingori for

his help over the years in the PATH project. My appreciation also goes to the rest of

my committee for their time and helpful comments in support of my dissertation.

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VITA

June 17, 1970 Born, Anderson, South Carolina

1992 B.S., Power Mechanical EngineeringNational Tsing Hua UniversityTaiwan

1994 M.S., Mechanical EngineeringUniversity of California, Los Angeles

1993-2000 Graduate Research Assistant

Mechanical and Aerospace Engineering DepartmentUniversity of California, Los Angeles

PUBLICATIONS

Robert H. Chen and Jason L. Speyer, A Generalized Least-Squares Fault DetectionFilter, to be published in the International Journal of Adaptive Control and

Signal Processing - Special Issue on Fault Detection and Isolation, 2000.

Robert H. Chen and Jason L. Speyer, Optimal Stochastic Multiple-Fault Detection

Filter, Proceedings of the 38th Conference on Decision and Control, 1999, pp.

4965-4970.

Robert H. Chen and Jason L. Speyer, Optimal Stochastic Fault Detection Filter,

Proceedings of the American Control Conference, 1999, pp. 91-96.

Robert H. Chen and Jason L. Speyer, Residual-Sensitive Fault Detection Filter,Proceedings of the 7th IEEE Mediterranean Conference on Control and Au-

tomation, 1999, pp. 835-851

Randal K. Douglas, Robert H. Chen and Jason L. Speyer,Model Input Reduction,

Proceedings of the American Control Conference, 1997, pp. 3882-3886.

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Randal K. Douglas, Durga P. Malladi, Robert H. Chen, D. Lewis Mingori and Jason

L. Speyer,Fault Detection and Identification for Advanced Vehicle Control Sys-

tems, Proceedings of the 13th World Congress, 1996, pp. 201-206.

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ABSTRACT OF THE DISSERTATION

Fault Detection Filters

for

Robust Analytical Redundancy

by

Robert Hsu Chen

Doctor of Philosophy in Mechanical Engineering

University of California, Los Angeles, 2000

Professor Jason L. Speyer, Chair

In this dissertation, three fault detection and identification algorithms are presented.

The first two design algorithms, called the generalized least-squares fault detection

filter and the optimal stochastic fault detection filter, are determined for the unknown

input observer. The objective of both filters is to monitor a single fault called the

target fault and block other faults which are called nuisance faults. The first filter is

derived from solving a min-max problem with a generalized least-squares cost criterion

which explicitly makes the residual sensitive to the target fault, but insensitive to the

nuisance faults. The second filter is derived by minimizing the transmission from

the nuisance faults to the projected output error while maximizing the transmission

from the target fault so that the residual is affected primarily by the target fault

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and minimally by the nuisance faults. It is shown that both filters approximate the

properties of the classical unknown input observer. Filter designs can be obtained for

both time-invariant and time-varying systems.

The third design algorithm, called the robust multiple-fault detection filter, is

determined for the restricted diagonal detection filter of which the Beard-Jones de-

tection filter is a special case. The filter is derived by dividing the output error into

several subspaces. For each subspace, the transmission from one fault is maximized,

and the transmission from other faults is minimized. Therefore, each projected resid-

ual is affected primarily by one fault and minimally by other faults. It is shown

that this filter approximates the properties of the classical restricted diagonal detec-

tion filter. This filter is different from other algorithms for the restricted diagonal or

Beard-Jones detection filter which explicitly force the geometric structure by using

eigenstructure assignment or geometric theory. Rather, this filter is derived from

solving an optimization problem and only in the limit, is the geometric structure of

the restricted diagonal detection filter recovered. When it is not in the limit, the

filter only isolates the faults within approximate invariant subspaces. This new fea-

ture allows the filter to be potentially more robust since the filter structure is lessconstrained. Filter designs can be obtained for both time-invariant and time-varying

systems.

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Chapter 1

Introduction

Any system under automatic control demands a high degree of reliability to operate

properly. If a fault develops in the plant, the controller will not work properly since

it is designed on the nominal plant. The controller also relies on the health of the

sensors and actuators. If a sensor fails, the controllers command will be generated

by using incorrect measurements. If an actuator fails, the controllers command will

not be applied properly to the plant. To avoid this situation, one needs a health

monitoring system capable of detecting a fault as it occurs and identifying the faulty

component. This process is called fault detection and identification.

The most common approach to fault detection and identification is hardware

redundancy which is the direct comparison of the output from identical components.

This approach requires very little computation. However, hardware redundancy is

expensive and limited by space and weight. An alternative is analytical redundancy

which uses the modeled dynamic relationship between system inputs and measured

system outputs to form a residual process used for detecting and identifying faults.

Nominally, the residual is nonzero only when a fault has occurred and is zero at

other times. Therefore, no redundant components are needed. However, additional

computation is required.

A popular approach to analytical redundancy is the detection filter which was

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first introduced by (Beard, 1971) and refined by (Jones, 1973). It is also known

as Beard-Jones detection (BJD) filter. A geometric interpretation and a spectral

analysis of the BJD filter are given in (Massoumnia, 1986) and (White and Speyer,

1987), respectively. The idea of the BJD filter is to place the reachable subspace of

each fault into invariant subspaces which do not overlap each other. Then, when a

nonzero residual is detected, a fault can be announced and identified by projecting

the residual onto each of the invariant subspaces. In this way, multiple faults can

be monitored in one filter. A design algorithm (Douglas and Speyer, 1999) improves

the robustness of the BJD filter by imposing the geometric structure to completely

isolate the faults, then the design freedom remaining is used to bound the process

and sensor noise transmission.

In (Massoumnia, 1986), a more general form of the detection filter, called the re-

stricted diagonal detection (RDD) filter, is given of which the BJD filter is a special

case. Instead of placing each fault into an invariant subspace like the BJD filter does,

the RDD filter places all the other faults associated with each fault, which needs to

be detected, into the unobservable subspace of a projected residual. Therefore, each

projected residual is only sensitive to one fault, but not to the other faults. Whenevery fault is detected, it is shown that the RDD filter is equivalent to the BJD

filter (Massoumnia, 1986). However, some faults do not need to be detected, but

only need to be blocked from the projected residuals. For example, certain process

noise and plant uncertainties may be modeled as faults. By relaxing the constraint

on detecting some faults which do not need to be detected, the RDD filter, which is

more general and more robust, is obtained (Douglas and Speyer, 1996). Note that

the design algorithms for the RDD or BJD filter, which rely on the eigenstructure

assignment (White and Speyer, 1987; Douglas and Speyer, 1996) or geometric the-

ory (Massoumnia, 1986; Douglas and Speyer, 1999), limit the applicability of the

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detection filter to time-invariant systems and have not enhanced the sensitivity of

the projected residuals to their associated faults from the filter structure.

One related approach, the unknown input observer (Massoumnia et al., 1989;

Patton and Chen, 1992), is another special case of the RDD filter when only onefault is detected. The faults are divided into two groups: a single target fault and

possibly several nuisance faults. The nuisance faults are placed in the unobservable

subspace of the residual. Therefore, the residual is sensitive to the target fault, but

not to the nuisance faults. Although only one fault can be monitored in each unknown

input observer, an approximate unknown input observer can be obtained by solving

a disturbance attenuation problem (Chung and Speyer, 1998). By adjusting the

disturbance attenuation bound, the filter can either completely block the nuisance

faults or partially block the nuisance faults. This new feature allows the filter to

be potentially more robust since the filter structure is less constrained. Another

benefit of the approximate unknown input observer is that time-varying systems can

be treated.

In Section 1.1, the background of the fault detection filter is given. In Section 1.2,

there is an overview of the dissertation.

1.1 Fault Detection Filter Background

In this section, the background of the fault detection filter is given. In Section 1.1.1,

the models of the plant, actuator and sensor faults are given (Beard, 1971; White

and Speyer, 1987; Chung and Speyer, 1998). In Sections 1.1.2 and 1.1.3, the BJD

filter problem and the RDD filter problem are stated, and the geometric solutions are

given, respectively (Massoumnia, 1986; Douglas, 1993). In Sections 1.1.4 and 1.1.5,

the unknown input observer problem and the approximate unknown input observer

problem are reviewed, respectively (Massoumnia et al., 1989; Chung and Speyer,

1998).

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1.1.1 Fault Modeling

In this section, the models of the plant, actuator and sensor faults are given (Beard,

1971; White and Speyer, 1987; Chung and Speyer, 1998). Consider a linear system,

x = Ax + Buu (1.1a)

y = Cx (1.1b)

where u is the control input and y is the measurement. System matrices A, Bu and

C can be time-varying. All system variables belong to real vector spaces.

For a fault in the ith actuator, it is modeled as an additive term in the state

equation (1.1a) (Beard, 1971; White and Speyer, 1987).

x = Ax + Buu + Faiai

where Fai is the ith column of Bu and ai is an unknown and arbitrary function of

time that is zero when there is no fault. The failure mode ai models the time-varying

amplitude of the actuator fault while the failure signature Fai models the directional

characteristics of the actuator fault. For example, a stuck ith actuator fault can be

modeled as ui + ai = ca where ui is the control command of the ith actuator and ca

is a constant. For a fault in the plant, it can be modeled similarly by pulling out the

corresponding entries in the A matrix.

For a fault in the ith sensor, it is modeled as an additive term in the measurement

equation (1.1b) (Beard, 1971; White and Speyer, 1987).

y = Cx + Esisi (1.2)

where Esi is a column of zeros except a one in the ith position and si is an unknown

and arbitrary function of time that is zero when there is no fault. The failure mode

si models the time-varying amplitude of the sensor fault while the failure signature

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Esi models the directional characteristics of the sensor fault. For example, a bias ith

sensor fault can be modeled as si = cs where cs is a constant.

For the fault detection filter design, an input to the plant which drives the mea-

surement in the same way that si does in (1.2) is obtained (Chung and Speyer,

1998). Define a new state x,

x = x + fsisi

where Esi = Cfsi. Then, (1.2) can be written as

y = Cx

and the dynamic equation of x is

x = Ax + Buu +

Afsifsi fsi si

si

(1.3)

Here fsi and si are assumed once continuously differentiable. Therefore, for the fault

detection filter design, the sensor fault is modeled as a two-dimension additive term

in the state equation (1.3). The interpretation of (1.3) is that Afsifsi represents

the sensor fault magnitude si direction and fsi represents the sensor fault rate si

direction. This suggests that a possible simplification when information about the

spectral content of the sensor fault is available. If it is known that the sensor fault

has persistent and significant high frequency components, the fault direction could

be approximated by the fsi direction. Or, if it is known that a sensor fault has only

low frequency components, such as in the case of a bias, the fault direction could be

approximated by the Afsi

fsi direction.

1.1.2 Beard-Jones Detection Filter

In this section, the BJD filter problem is reviewed by using geometry theory (Mas-

soumnia, 1986; Douglas, 1993). Following the development in Section 1.1.1, any

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plant, actuator and sensor fault can be modeled as an additive term in the state

equation. Therefore, a linear time-invariant observable system with q failure modes

can be modeled by

x = Ax + Buu +

qi=1

Fii (1.4a)

y = Cx (1.4b)

Assume the Fi are monic so that i = 0 imply Fii = 0.Consider a linear observer,

x = Ax + Buu + L(y Cx) (1.5)

and the residual

r = y Cx

By using (1.4) and (1.5), the dynamic equation of the error, e = x x, is

e = (A LC)e +q

i=1

Fii

and the residual can be written as

r = Ce

If the observer gain L is chosen to make A LC stable. After the transient responsedue to the initial condition error, the residual is nonzero only if a failure mode i is

nonzero and is almost always nonzero whenever i is nonzero. Therefore, any stable

observer can detect the occurrence of a fault by monitoring the residual and when

it is nonzero, a fault has occurred. A more difficult task is to determine which fault

has occurred and that is what a fault detection filter is designed to do.

The objective of the BJD filter problem is to choose a filter gain L such that when

a fault i occurs, the error remains in a (C, A)-invariant subspace which contains the

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reachable subspace of (A LC,Fi). Thus, the residual remains in a fixed outputsubspace. Furthermore, the output subspace for each fault is independent to each

other. Then, the fault can be identified by projecting the residual onto each of the

output subspaces. From geometric point of view, the (C, A)-invariant subspaces form

the structure of the BJD filter while the filter gain L provides little or no insight into

this structure. Furthermore, given a set of (C, A)-invariant subspaces, L is not hard

to find. Therefore, the (C, A)-invariant subspaces will be discussed instead of L.

The minimal (C, A)-invariant subspace of Fi, called Wi, is given by the recursivealgorithm (Wonham, 1985)

W0i = (1.6)

Wk+1i = ImFi A(Wki

Ker C) (1.7)

For dim Fi = 1, the recursive algorithm implies

Wi =

Fi AFi AkiFi

where ki is the smallest non-negative integer such that CAkiFi = 0. If the (C, A)-

invariant subspace for each fault is chosen as Wi, the invariant zeros of (C,A,Fi) willbecome part of the eigenvalues of the BJD filter (Massoumnia, 1986).

To avoid this situation, the (C, A)-invariant subspace for each fault is chosen as

Ti = Wi Vi (1.8)

where Vi is the subspace spanned by the invariant zero directions of (C,A,Fi). Then,the invariant zeros of (C,A,Fi) will not become part of the filter eigenvalues (Mas-

soumnia, 1986). Ti is called the detection space or the minimal (C, A)-unobservabilitysubspace of Fi because it is the unobservable subspace of (HiC, A LC) where

Hi : Y Y , Ker Hi = CTi , Hi = I CTi[(CTi)TCTi]1(CTi)T

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for some filter gains L. Note that CTi = CWi because CVi = 0.Given that the (C, A)-invariant subspace for each fault is chosen as Ti, the neces-

sary and sufficient condition of the existence of the filter gain L is that F1 Fq are

output separable and mutually detectable (Massoumnia, 1986). F1 Fq are outputseparable if

CTi j=i

CTj = 0

Output separability ensures that each fault can be isolated from other faults. When

a fault i occurs, the error remains in the subspace Ti and the residual remains inthe output subspace C

Ti. IfC

T1

C

Tq are independent, the fault can be identified

by projecting the residual onto each CTi. If the faults are not output separable,then usually, the designers needs to discard some faults from the design set. Output

separability also implies that the projected residuals will be nonzero for at least a

period of time when their associated faults occur (Chung and Speyer, 1998).

F1 Fq are mutually detectable if ( C, A, [ F1 Fq ]) does not have more invari-ant zeros than (C,A,Fi), i = 1 q. Mutual detectability ensures that every eigen-

value of the BJD filter can be assigned. If the faults are not mutually detectable, the

extra invariant zeros will become part of the filter eigenvalues. If the extra invariant

zeros are in the right-half plane, no stable BJD filter can be obtained.

It is desired that the projected residuals remain nonzero as long as their associated

faults exist. For a bias fault i, the steady-state residual is zero if (C, A LC,Fi) hasinvariant zeros at origin (Kwakernaak and Sivan, 1972a). Since the filter gain L does

not change the invariant zero, (C, A

LC,Fi) has invariant zeros at origin if and only

if (C,A,Fi) has invariant zeros at origin. Therefore, to ensure a nonzero projected

residual in steady state when its associated fault occurs, (C,A,Fi), i = 1 q, do nothave invariant zeros at origin.

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By summarizing the results above, there are three assumptions about the system

(1.4) that are needed in order to have a well-conditioned BJD filter.

Assumption 1.1. F1

Fq are output separable.

Assumption 1.2. F1 Fq are mutually detectable.

Assumption 1.3. (C,A,Fi), i = 1 q, do not have invariant zeros at origin.

There are several design algorithms developed for the BJD filter to determine the

filter gain (White and Speyer, 1987; Douglas and Speyer, 1996, 1999). In (White

and Speyer, 1987), an eigenstructure assignment algorithm, which places the right

eigenvectors of the BJD filter to span the minimal (C, A)-unobservability subspace

of each fault, is presented. In (Douglas and Speyer, 1996), an eigenstructure assign-

ment algorithm, which places the left eigenvectors of the BJD filter to annihilate the

minimal (C, A)-unobservability subspace of each fault, is presented. Note that these

two design algorithms assign the filter eigenvalues arbitrarily and have not considered

any disturbance. In (Douglas and Speyer, 1999), the robustness of the BJD filter is

improved by imposing the geometric structure to completely isolate the faults, then

the design freedom remaining is used to bound the H norm of the transfer matrixfrom the process and sensor noise to the projected residuals. After the filter gain has

been determined, the sensitivity of the projected residuals to their associated faults is

enhanced. Each projected residual is modified by multiplying a constant row vector

from the left. Then, the ratio of the H norm of the transfer matrix from each faultto its associated modified projected residual to the

H norm of the transfer matrix

from the process and sensor noise to each modified projected residual is maximized

with respect to this vector. Note that the filter structure is not used to enhance the

sensitivity of the projected residuals to their associated faults.

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After the filter gain has been determined, the error remains in Ti and the residualremains in CTi when a fault i occurs. Therefore, the fault can be identified byprojecting the residual onto each CTi. This is done by using projectors Hi, i = 1 q,

which annihilate [ CT1 CTi1 CTi+1 CTq ]

= CTi.

Hi : Y Y , Ker Hi = CTi , Hi = I CTi[(CTi)TCTi]1(CTi)T (1.9)

The projected residual Hir is nonzero only when the fault i is nonzero and is zero

even if other faults j=i are nonzero. Therefore, by monitoring Hir, i = 1 q, everyfault can be detected and identified.

1.1.3 Restricted Diagonal Detection Filter

In this section, the RDD filter problem is reviewed by using geometry theory (Mas-

soumnia, 1986; Douglas, 1993). The RDD filter is a more general form of the detection

filter of which the BJD filter is a special case.

From Section 1.1.2, the dynamic equation of the error is

e = (A LC)e +q

i=1Fii

and the projected residuals are

Hir = HiCe

where i = 1 q. If the filter gain L is chosen to make the unobservable subspace of(HiC, A LC) contains the reachable subspace of (A LC, Fi) where Fi = [ F1 Fi1 Fi+1 Fq ]. Then, the projected residual Hir is only sensitive to the faulti, but not to the other faults [ T1 Ti1 Ti+1 Tq ]T = i. Therefore, instead ofplacing each i into its minimal (C, A)-unobservability subspace Ti like the BJD filterdoes, the RDD filter places each i into its minimal (C, A)-unobservability subspace

Ti if i needs to be detected (Massoumnia, 1986).

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When every fault is detected, it is shown that the RDD filter is equivalent to the

BJD filter (Massoumnia, 1986). However, some faults do not need to be detected, but

only need to be blocked from the projected residuals. For example, certain process

noise and plant uncertainties may be modeled as faults. Therefore, the RDD filter is

a more general form of the detection filter because it does not require every fault to

be detected while the BJD filter does. By relaxing the constraint on detecting some

faults which do not need to be detected, the RDD filter is more robust than the BJD

filter (Douglas and Speyer, 1996).

There are three assumptions about the system (1.4) that are needed in order to

have a well-conditioned RDD filter (Massoumnia, 1986). Assumption 1.4 ensures that

each fault can be isolated from other faults. Assumption 1.5 ensures that every eigen-

value of the RDD filter can be assigned. Assumption 1.6 ensures a nonzero projected

residual in steady state when its associated fault occurs. Note that Assumption 1.6

is less restrict than Assumption 1.3.

Assumption 1.4. F1 Fq are output separable.

Assumption 1.5. F1 Fq are mutually detectable.

Assumption 1.6. (C,A,Fi) does not have invariant zeros at origin if i needs to

be detected.

The only design algorithm (Douglas and Speyer, 1996) for the RDD filter is an

eigenstructure assignment algorithm which places the left eigenvectors of the RDD

filter to annihilate the minimal (C, A)-unobservability subspace of each fault. Note

that this design algorithm assigns the filter eigenvalues arbitrarily and has not con-

sidered any disturbance.

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1.1.4 Unknown Input Observer

In this section, the unknown input observer problem is reviewed (Massoumnia et al.,

1989). The unknown input observer is another special case of the RDD filter when

only one fault is detected. The faults are divided into two groups: a single target

fault and possibly several nuisance faults. The nuisance faults are placed in the

unobservable subspace of the projected residual. Therefore, the projected residual is

sensitive to the target fault, but not to the nuisance faults. Note that there is only

one projected residual because only the target fault needs to be detected.

Consider a linear time-invariant observable system with q failure modes,

x = Ax + Buu +

qi=1

Fii (1.10a)

y = Cx (1.10b)

Assume the Fi are monic so that i = 0 imply Fii = 0. Since the unknown inputobserver is designed to detect only one fault and not to be affected by other faults,

let 1 = i be the target fault and 2 = [ T1 Ti1 Ti+1 Tq ]T be the nuisance

fault. Then, (1.10) can be rewritten as

x = Ax + Buu + F11 + F22 (1.11a)

y = Cx (1.11b)

where F1 = Fi and F2 = [ F1 Fi1 Fi+1 Fq ].There are two assumptions about the system (1.11) that are needed in order

to have a well-conditioned unknown input observer (Massoumnia et al., 1989). As-

sumption 1.7 ensures that the target fault can be isolated from the nuisance fault.

Assumption 1.8 ensures a nonzero projected residual in steady state when the target

fault occurs.

Assumption 1.7. F1 and F2 are output separable.

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Assumption 1.8. (C,A,F1) does not have invariant zeros at origin.

Note that the unknown input observer problem does not need the mutual de-

tectability assumption because only one minimal (C, A)-unobservability subspace is

formed for the nuisance fault. There is no minimal (C, A)-unobservability subspace

formed for the target fault.

Since the unknown input observer only places the nuisance fault into its mini-

mal (C, A)-unobservability subspace T2 while leaves the target fault unrestricted, theunknown input observer is a special case of the RDD and BJD filters. When only

one fault is detected, the RDD filter is equivalent to the unknown input observer.

When only one fault is considered, the BJD filter is equivalent to the unknown input

observer. Therefore, the design algorithms (White and Speyer, 1987; Douglas and

Speyer, 1996, 1999) for the RDD or BJD filter can be used to determined the filter

gain of the unknown input observer. Note that these design algorithms, which rely on

the eigenstructure assignment (White and Speyer, 1987; Douglas and Speyer, 1996)

or geometric theory (Douglas and Speyer, 1999), limit the applicability of the fault

detection filter to time-invariant systems and have not enhanced the sensitivity of

the projected residuals to their associated faults from the filter structure.

After the filter gain has been determined, the error remains in T2 and the residualremains in CT2 when the nuisance fault occurs. When the target fault occurs, theerror and residual are not in some particular subspaces because there is no minimal

(C, A)-unobservability subspace formed for the target fault. Therefore, the projected

residual Hr is only sensitive to the target fault, but not to the nuisance fault where

H : Y Y , Ker H = CT2 , H = I CT2[(CT2)T

CT2]1

(CT2)T

(1.12)

The projected residual Hr is nonzero only when the target fault is nonzero and is

zero even if the nuisance fault is nonzero. Therefore, by monitoring Hr, the target

fault can be detected.

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1.1.5 Approximate Unknown Input Observer

In this section, the approximate unknown input observer problem is reviewed (Chung

and Speyer, 1998). Although only one fault can be monitored in each unknown input

observer, an approximate unknown input observer is obtained by solving a distur-

bance attenuation problem (Chung and Speyer, 1998). By adjusting the disturbance

attenuation bound, the filter can either completely block the nuisance fault or par-

tially block the nuisance fault. This new feature allows the filter to be potentially

more robust since the filter structure is less constrained. Another benefit of the

approximate unknown input observer is that time-varying systems can be treated.

Consider a linear system similar to (1.11),

x = Ax + Buu + F11 + F22 (1.13a)

y = Cx (1.13b)

where the system matrices A, Bu, C, F1 and F2 can be time-varying. There are

three assumptions about the system (1.13) that are needed in order to have a well-

conditioned unknown input observer. Assumption 1.9 is the general requirement to

design any linear observer (Kwakernaak and Sivan, 1972a). Assumption 1.10 ensures

that the target fault can be isolated from the nuisance fault (Chung and Speyer, 1998).

Assumption 1.11 ensures for time-invariant systems, a nonzero projected residual in

steady state when the target fault occurs.

Assumption 1.9. For time-varying systems, (C, A) is uniformly observable. For

time-invariant systems, (C, A) is detectable.

Assumption 1.10. F1 Fq are output separable.

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Assumption 1.11. For time-invariant systems, (C,A,F1) does not have invariant

zeros at origin.

The output separability test is

Rank

CT1 CT2

= p1 +p2 (1.14)

where p1 = dim F1 and p2 = dim F2. For time-invariant systems (Massoumnia et al.,

1989),

CTi =

CAi,1fi,1 CAi,pi fi,pi

(1.15)

The vector fi,j , i = 1 and 2, j = 1 pi, is the j-th column of Fi. i,j is the smallestnon-negative integer such that CAi,jfi,j = 0. For time-varying systems (Chung andSpeyer, 1998),

CTi =

C(t)bi,1,i,1(t) C(t)bi,pi,i,pi(t)

(1.16)

The vectors bi,j,i,j(t), i = 1 and 2, j = 1 pi, are found from the iteration definedby the Goh transformation (Bell and Jacobsen, 1975),

bi,j,0(t) = fi,j(t)

bi,j,k(t) = A(t)bi,j,k1(t) bi,j,k1(t)

where fi,j(t) is the j-th column of Fi. i,j is the smallest non-negative integer such

that C(t)bi,j,i,j(t) = 0 for t [t0, t1].

Remark 1.The output separability test (1.14) is based on the assumption that the

vectors in F1 and F2 are output separable, respectively, i.e.,

Rank CTi = pi

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where i = 1 and 2. If the vectors in either F1 or F2 are not output separable, a new

basis for F1 or F2 can be found such that the vectors in F1 and F2 are output separable,

respectively. This will be discussed in Section 2.4.

The minimal (C, A)-invariant subspace of Fi is

Wi =

fi,1 Ai,1fi,1 fi,2 Ai,2fi,2

for time-invariant systems (Massoumnia et al., 1989) and

Wi =

bi,1,0(t) bi,1,i,1(t) bi,2,0(t) bi,2,i,2(t)

for time-varying systems (Chung and Speyer, 1998). However, the minimal (C, A)-

unobservability subspace of Fi can not be determined by (1.8) for time-varying

systems because the idea of the invariant zero direction is only defined for time-

invariant systems. A (C, A)-invariant subspace, which is similar to the minimal

(C, A)-unobservability subspace, will be introduced for time-varying systems in Chap-

ter 2.

The first design algorithm for the approximate unknown input observer is (Chung

and Speyer, 1998). Since two other design algorithms will be presented in Chapters 2

and 3, the filter gain determination will be explained later. After the filter gain has

been determined, the residual remains in CT2, (1.15) or (1.16), when the nuisancefault occurs. Therefore, by monitoring the projected residual Hr where H is (1.12)

subject to (1.15) or (1.16), the target fault can be detected.

1.2 Overview of the Dissertation

In this dissertation, three fault detection and identification algorithms are presented.

The first two design algorithms, called the generalized least-squares fault detection

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filter and the optimal stochastic fault detection filter, are determined for the ap-

proximate unknown input observer. The third design algorithm, called the robust

multiple-fault detection filter, is determined for the approximate RDD filter.

In Chapter 2, the generalized least-squares fault detection filter, motivated by

(Bryson and Ho, 1975; Chung and Speyer, 1998), is presented. A new least-squares

problem with an indefinite cost criterion is formulated as a min-max problem by

generalizing the least-squares derivation of the Kalman filter (Bryson and Ho, 1975)

and allowing the explicit dependence on the target fault which is not presented in

(Chung and Speyer, 1998). Since the filter is derived similarly to (Chung and Speyer,

1998), many properties obtained in (Chung and Speyer, 1998) also apply to this

filter. However, some new important properties are given. For example, since the

target fault direction is now explicitly in the filter gain calculation, a mechanism

is provided which enhances the sensitivity of the projected residual to the target

fault. Furthermore, the projector, which annihilates the residual direction associated

with the nuisance faults and is assumed in the problem formulation of (Chung and

Speyer, 1998), is not required in the derivation of this filter. Finally, the nuisance fault

directions are generalized for time-invariant systems so that their associated invariantzero directions are included in the invariant subspace generated by the filter. This

prevents the associated invariant zeros from becoming part of the eigenvalues of the

filter. It is also shown that this filter completely blocks the nuisance faults in the

limit where the weighting on the nuisance faults is zero. In the limit, the nuisance

faults are placed in a minimal (C, A)-unobservability subspace for time-invariant

systems and a similar invariant subspace for time-varying systems. Therefore, the

generalized least-squares fault detection filter becomes equivalent to the unknown

input observer in the limit and extends the unknown input observer to the time-

varying case. Reduced-order filters are derived in the limit for both time-invariant

and time-varying systems.

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In Chapter 3, extensions and analysis of a fault detection filter, first developed

in (Lee, 1994; Brinsmead et al., 1997), are presented. This filter, called optimal

stochastic fault detection filter, is derived by minimizing the transmission from the

nuisance faults while maximizing the transmission from the target fault. The trans-

mission, which is generalized to a vector, is defined on the output error by using a

matrix projector derived from solving the optimization problem. Some new proper-

ties of this filter in the limit where the weighting on the nuisance fault transmission

goes to infinity are given. It is shown that this filter completely blocks the nuisance

faults in the limit by placing them into a minimal (C, A)-unobservability subspace

for time-invariant systems and a similar invariant subspace for time-varying systems.

Therefore, the optimal stochastic fault detection filter recovers the unknown input

observer in the limit and extends the unknown input observer to the time-varying

case.

In Chapter 4, the robust multiple-fault detection filter is presented. The filter

is derived by dividing the output error into several subspaces. For each subspace,

the transmission from one fault, denoted the associated target fault, is maximized,

and the transmission from other faults, denoted the associated nuisance fault, isminimized. Therefore, each projected residual is affected primarily by one fault and

minimally by other faults. The cost criterion is constructed such that the output

error variance due to each associated target fault is to be maximized and the output

error variance due to each associated nuisance fault, process noise, sensor noise and

initial conditional error is to be minimized with respect to the filter gain and the

projectors used for dividing the output error. Therefore, each associated target and

nuisance faults are included in the cost criterion, in turn, as a sum which produces

approximately the geometric structure of the RDD filter.

For both time-invariant and time-varying systems, it is shown that, in the limit

where the weighting on each associated nuisance fault transmission goes to infinity,

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the robust multiple-fault detection filter places each associated nuisance fault into

the unobservable subspace of the associated projected residual when there is no com-

plementary subspace.1 Therefore, the robust multiple-fault detection filter becomes

equivalent to the RDD filter in the limit and extends the RDD filter to the time-

varying case. Numerical examples show that the filter is an approximate RDD filter

when it is not in the limit even if there exists the complementary subspace. These

limiting results are important in ensuring that both fault detection and identification

can occur.

This filter is different from the only other existing algorithm (Douglas and Speyer,

1996) for the RDD filter which explicitly forces the geometric structure by using

eigenstructure assignment. Rather, this filter is derived from solving an optimization

problem and only in the limit, is the geometric structure of the RDD filter recovered.

When it is not in the limit, the filter only isolates the faults within approximate un-

observable subspaces. This new feature allows the filter to be potentially more robust

since the filter structure is less constrained. Furthermore, the filter can be applied to

time-varying systems since it is derived from solving an optimization problem which

also allows the presence of process and sensor noise. Finally, since the associatedtarget fault directions are explicitly in the filter gain calculation, a mechanism is

provided which enhances the sensitivity of the projected residuals to their associated

target faults. Nevertheless, this unique optimization problem allows the design of the

detection filter in its most general and potentially most robust form: an approximate

RDD filter. Note that the eigenstructure assignment approach (Douglas and Speyer,

1996), which only applies to time-invariant systems, assigns the filter eigenvalues ar-

bitrarily and does not consider any disturbance nor enhance the sensitivity of the

projected residuals to their associated target faults. Although this new filter has all

1The union of the (C,A)-invariant subspace of each fault is assumed to fill the entire state spaceleaving no remaining subspace, the complementary subspace.

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these advantages, the process of deriving the filter gain requires the solution to a two-

point boundary value problem which includes a set of Lyapunov equations. However,

the filter gain computation can be done off-line so that the filter implementation is

as straightforward as the RDD filter.

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Chapter 2

A Generalized Least-Squares Fault

Detection Filter

In this chapter, a fault detection and identification algorithm is determined from a

generalization of the least-squares derivation of the Kalman filter. The objective of

the filter is to monitor a single fault called the target fault and block other faults which

are called nuisance faults. The filter is derived from solving a min-max problem with

a generalized least-squares cost criterion which explicitly makes the residual sensitive

to the target fault, but insensitive to the nuisance faults. It is shown that this filter

approximates the properties of the classical fault detection filter such that in the limit

where the weighting on the nuisance faults is zero, the generalized least-squares fault

detection filter becomes equivalent to the unknown input observer where there exists

a reduced-order filter. However, the nuisance fault directions and their associated

invariant zero directions must be included in the invariant subspace generated by

the generalized least-squares fault detection filter. Filter designs can be obtained for

both time-invariant and time-varying systems.

The problem is formulated in Section 2.1 and its solution is derived in Section 2.2(Chung and Speyer, 1998; Bryson and Ho, 1975; Rhee and Speyer, 1991; Banavar and

Speyer, 1991). In Section 2.3, some conditions for this problem are derived by using

linear matrix inequality (Chung and Speyer, 1998). In Section 2.4, the filter is derived

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in the limit (Chung and Speyer, 1998; Bell and Jacobsen, 1975). In Section 2.5, it is

shown that, in the limit, the nuisance faults are placed in an invariant subspace. For

time-invariant systems, this subspace is the minimal (C, A)-unobservability subspace.

In Section 2.6, reduced-order filters are derived in the limit for both time-invariant

and time-varying systems. In Section 2.7, numerical examples are given.

2.1 Problem Formulation

Consider a linear system,

x = Ax + Buu + Bww (2.1a)

y = Cx + v (2.1b)

where u is the control input, y is the measurement, w is the process noise, and v is the

sensor noise. System matrices A, Bu, Bw and C are time-varying and continuously

differentiable. All system variables belong to real vector spaces, x X, u Uandy Y.

Following the development in Section 1.1.1, any plant, actuator, and sensor fault

can be modeled as an additive term in the state equation (2.1a). Therefore, a linear

system with q failure modes can be modeled by

x = Ax + Buu + Bww +

qi=1

Fii (2.2a)

y = Cx + v (2.2b)

where i belong to real vector spaces, and Fi are time-varying and continuously

differentiable. Assume the Fi are monic so that i = 0 imply Fii = 0. Sincethe generalized least-squares fault detection filter is designed to detect only one

fault and not to be affected by other faults, let 1 = i be the target fault and

2 = [ T1 Ti1 Ti+1 Tq ]T be the nuisance fault. Then, (2.2) can be rewritten

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as

x = Ax + Buu + Bww + F11 + F22 (2.3a)

y = Cx + v (2.3b)

where F1 = Fi and F2 = [ F1 Fi1 Fi+1 Fq ].There are three assumptions about the system (2.3) that are needed in order

to have a well-conditioned unknown input observer. Assumption 2.1 is the gen-

eral requirement to design any linear observer (Kwakernaak and Sivan, 1972a). As-

sumption 2.2 ensures that the target fault can be isolated from the nuisance fault

(Massoumnia et al., 1989; Chung and Speyer, 1998). Assumption 2.3 ensures for

time-invariant systems, a nonzero residual in steady-state when the target fault oc-

curs.

Assumption 2.1. For time-varying systems, (C, A) is uniformly observable. For

time-invariant systems, (C, A) is detectable.

Assumption 2.2. F1 and F2 are output separable.

Assumption 2.3. For time-invariant systems, (C,A,F1) does not have invariant

zeros at origin.

The objective of blocking the nuisance fault while detecting the target fault can

be achieved by solving the following min-max problem,

min1

max2

maxw

maxx(t0)

J (2.4)

where a generalized least-squares cost criterion is

J =1

2

tt0

1 2Q1

1

2 2Q12

w 2Q1w

y Cx 2V1

d

12

x(t0) x0 20 (2.5)

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subject to (2.3a). Note that, without the minimization with respect to 1, (2.5)

reduces to the standard least-squares cost criterion of the Kalman filter (Bryson

and Ho, 1975). t is the current time and y is assumed given. Q1, Q2, Qw, V and

0 are positive definite. is a non-negative scalar. Note that Q1, Q2, Qw, 0

and are design parameters to be chosen while V may be physically related to

the power spectral density of the sensor noise because of (2.3b) (Bryson and Ho,

1975). The interpretation of the min-max problem is the following. Let 1, 2, w

and x(t0) be the optimal strategies for 1, 2, w and x(t0), respectively. Then,

x(|Yt), the x associated with 1, 2, w and x(t0), is the optimal trajectory forx where

[t0, t] and given the measurement history Yt =

{y()

|t0

t

}. Since

1 maximizes y Cx and 2, w, x(t0) minimize y Cx, y Cx is made primarilysensitive to 1 and minimally sensitive to 2, w and x(t0). However, since x

is the

smoothed estimate of the state, a filtered estimate of the state, called x, is needed

for implementation. From the boundary condition in Section 2.2, at the current time

t, x(t|Yt) = x(t). Therefore, y Cx is primarily sensitive to the target fault andminimally sensitive to the nuisance fault, process noise and initial condition. Note

that when Q1 is larger, y Cx is more sensitive to the target fault. When issmaller, y Cx is less sensitive to the nuisance fault. In (Chung and Speyer, 1998),the cost criterion blocks the nuisance fault, but does not enhance the sensitivity to

the target fault. In Section 2.5, it is shown that the filter completely blocks the

nuisance fault when is zero by placing it into an invariant subspace, called Ker S.

Therefore, the residual used for detecting the target fault is

r = H(y Cx) (2.6)

where x, the filtered estimate of the state, is given in Section 2.2 and

H:Y Y, Ker H= CKerS , H=ICKerS[(CKerS)TCKerS]1(CKerS)T (2.7)

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Ker S is given and discussed in Sections 2.4 and 2.5.

Remark 2. The process noise can be considered as part of the nuisance fault

so that it could be completely blocked from the residual. However, the size of

the nuisance fault is limited because the target fault and nuisance fault have to

be output separable. Therefore, it is not always possible to include every pro-

cess noise to the nuisance fault. The plant uncertainties can also be considered

similarly to the process noise. In (Chung and Speyer, 1998), the process noise

and plant uncertainties can only be considered as part of the nuisance fault.

Remark 3. The differential game solved for the game-theoretic fault detection filter

(Chung and Speyer, 1998) is

minx

max2

maxy

maxx(t0)

J

where

J =

t1t0

HC(x x) 2Q

2 2M1 y Cx 2V1 dt x(t0) x0 2P10

subject to

x = Ax + Bu + F22

Note that the target fault is not included in the cost criterion nor the system. Also,

the derivation of the filter depends on the projector H which is defined apriori.

However, there is no need for the generalized least-squares fault detection filter to

explicitly introduce a projector in the cost criterion.

2.2 Solution

In this section, the min-max problem given by (2.4) is solved (Chung and Speyer,

1998; Bryson and Ho, 1975; Rhee and Speyer, 1991; Banavar and Speyer, 1991). The

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variational Hamiltonian of the problem is defined as

H = 12

1 2Q1

1

2 2Q12

w 2Q1w

y Cx 2V1

+ T(Ax + Buu + Bww + F11 + F22)

where Rn is a continuously differentiable Lagrange multiplier. The first-ordernecessary conditions (Bryson and Ho, 1975) imply that the optimal strategies for 1,

2, w and the dynamics of are

1 = Q1FT1 (2.8a)

2 =1

Q2F

T2 (2.8b)

w

= QwB

T

w (2.8c) = AT CTV1(y Cx) (2.8d)

with boundary conditions,

(t0) = 0[x(t0) x0] (2.8e)

(t) = 0 (2.8f)

By substituting (2.8a), (2.8b) and (2.8c) into (2.3a) and combining with (2.8d), thetwo-point boundary value problem requires the solution to

x

=

A 1

F2Q2F

T2 F1Q1FT1 +BwQwBTw

CTV1C AT

x

+

Buu

CTV1y

(2.9)

with boundary conditions (2.8e) and (2.8f). Note that x is now the state using the

optimal strategies (2.8a), (2.8b) and (2.8c). The form of (2.8e) suggests that

= (x x) (2.10)

where

(t0) = 0 (2.11a)

x(t0) = x0 (2.11b)

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and x is an intermediate state. By differentiating (2.10) and using (2.9),

0 =

+ A + AT +

1

F2Q2F

T2 F1Q1FT1 + BwQwBTw

CTV1C

x

+ AT + 1

F2Q2F

T

2 F1Q1FT

1 + BwQwBT

w

x

x + Buu + CTV1y

By adding and subtracting Ax and CTV1Cx,

0 =

+ A + AT +

1

F2Q2F

T2 F1Q1FT1 +BwQwBTw

CTV1C

(xx)

x + Ax + Buu + CTV1(y Cx)

Therefore, (2.10) is a solution to (2.9) if

x = Ax + Buu + CTV1(y Cx) (2.12)

= A + AT +

1

F2Q2F


CTV1C (2.13)

subject to (2.11).

By substituting 1 (2.8a), 2 (2.8b), w

(2.8c) and (2.10) into the cost criterion

(2.5),

J =1

2

tt0

x x 2

( 1F2Q2FT2 F1Q1FT1 +BwQwBTw) y Cx 2V1

d

12

x(t0) x0 20

By adding the zero term

0 =1

2 x(t0) x0 2(t0)

1

2 x(t) x(t) 2(t) +

1

2

tt0

d

d x x 2 d

to J,

J =1

2

tt0

x x 2

( 1F2Q2FT2 F1Q1FT1 +BwQwBTw) y Cx 2V1

+(x x)T(xx) + (xx)T(xx) + (xx)T(x x)

d

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Note that x(t) x(t) 2(t)= 0 because of the boundary condition (2.8f). By sub-stituting x (2.9), (2.10), (2.12), (2.13) into J and expanding y Cx 2V1 into (y Cx) C(x x) 2V1 ,

J = 12

t

t0

y Cx 2V1 d

Since x = x at current time t (2.8f), the generalized least-squares fault detection

filter is (2.12). Note that (2.12) is used by the residual (2.6) to detect the target

fault.

Remark 4. For a steady-state filter, (2.13) becomes

0 = A + AT +

1

F2Q2F


CTV1C (2.14)

The stability of the filter depends on showing that xTx is a Lyapunov function where

the coefficient matrix in the estimator equation (2.12) is Acl= A 1CTV1C. By

substituting Acl into (2.14),

Acl + ATcl =

1

F2Q2F


CTV1C

When Q1 = 0, given that (A, [ F2 Bw ]) is uniformly controllable for time-varying

systems and stabilizable for time-invariant systems, Acl + ATcl 0 and the filter

is exponentially stable for time-varying systems and asymptotically stable for time-

invariant systems (Kwakernaak and Sivan, 1972a). However, when Q1 = 0, Acl+ATcl might become indefinite. This can be interpreted as an attempt to make the

residual sensitive to the target fault. If Q1 is too large, the target fault could desta-

bilize the filter. If (A, [ F2 Bw ]) is not stabilizable, model reduction can be used to

remove the uncontrollable and unstable subspace (Moore, 1981). Note that the game-

theoretic fault detection filter (Chung and Speyer, 1998) is always stable because the

target fault is not in the problem formulation.

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2.3 Conditions for the Nonpositivity of the Cost

Criterion

In this section, the cost criterion (2.5) is converted into an equivalent linear matrix

inequality from which the sufficient conditions for optimality can be derived (Chung

and Speyer, 1998). The linear matrix inequality, associated with the solution opti-

mality, is just the left half of the saddle point inequality,

J(1, 2, w , x(t0)) J(1, 2, w, x(t0)) = 0 J(1, 2, w, x(t0))

The asterisk indicates that the optimal strategy is being used for that element.

By using a Lagrange multiplier (x

x)T to adjoin the constraint (2.3a) to the

cost criterion (2.5) and substituting 1 (2.8a),

J =1

2

tt0

2 2Q1

2

w 2Q1w

y Cx 2V1 +(x x)T

(Ax + Buu + Bww + F22 x)] d 12 x(t0) x0 20

By adding and subtracting 12tt0

(x x)TAxd and 12tt0

(x x)T xd to J,

J =1

2tt0 x x 2A 2 2Q12 w 2Q1w y Cx 2V1 +(x x)T

( x + Ax + Buu + Bww + F22) (x x)T(x x)

d

12

x(t0) x0 20

By integrating (x x)T(x x) by parts, substituting (2.3a) and (2.8a), adding andsubtracting 1

2

tt0

xTAT (x x)d to J,

J =1

2

t

t0

x

x

2+A+ATF1Q1FT1

2

2Q1

2

w

2Q1w

y

Cx

2V1

+ (x x)T( x + Ax + Buu + Bww + F22)

+( x + Ax + Buu + Bww + F22)T(x x)

d

12 x(t0) x0 20(t0)

1

2 x(t) x(t) 2(t)

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By expanding y Cx 2V1 into (y Cx) C(x x) 2V1 and substituting thegeneralized least-squares fault detection filter (2.12) into J,

J =1

2tt0 (x x)T wT T2 W

x xw

2 y Cx

2

V1 d

12

x(t0) x0 20(t0) 1

2 x(t) x(t) 2(t)

where

W =

+ A + AT F1Q1FT1 CTV1C Bw F2BTw Q1w 0

FT2 0 Q12

Therefore, the sufficient conditions for J

0 are

W 0 (2.15)

0 (t0) 0

(t) 0

In the limit where is zero, (2.15) becomes

F2 = 0 (2.16a)

+ A + AT + (F1Q1FT1 + BwQwBTw) CTV1C 0 (2.16b)

More detail about the limit is discussed in next section.

2.4 Limiting Case

In this section, the min-max problem (2.4) is solved in the limit where is zero (Chung

and Speyer, 1998; Bell and Jacobsen, 1975). It is shown that the solution satisfies

the sufficient condition (2.16) derived from the linear matrix inequality. When is

zero, there is no constraint on 2 to minimize y Cx. Therefore, the nuisance faultis completely blocked from the residual which is shown in Sections 2.5 and 2.6.

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In the limit, the cost criterion (2.5) becomes

J =1

2

tt0

1 2Q1

1

w 2Q1w

y Cx 2V1

d 12 x(t0) x0 20 (2.17)

This problem is singular with respect to 2. Therefore, the Goh transformation (Bell

and Jacobsen, 1975) is used to form a nonsingular problem. Let

1() =

t0

2(s)ds

1 = x F21 (2.18)

By differentiating (2.18) and using (2.3a),

1 = A1 + Buu + Bww + F11 + B11 (2.19)

where B1 = AF2 F2. By substituting (2.18) into the limiting cost criterion (2.17),

J =1

2

tt0

1 2Q1

1

w 2Q1w

1 2FT2CTV1CF2

y C1 2V1+(y C1)TV1CF21 + T1 FT2 CTV1(y C1)

d

12

1(t+0 ) + F21(t+0 ) x0 20 (2.20)

Then, the new min-max problem is

min1

max1

maxw

max1(t

+0)

J (2.21)

subject to (2.19).

If FT2 CTV1CF2 fails to be positive definite, (2.21) is still a singular problem

with respect to 1. Then, the Goh transformation has to be used until the problem

becomes nonsingular. If FT

2 CT

V1

CF2 = 0, let

2() =

t0

1(s)ds

2 = 1 B12 (2.22)

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Then,

2 = A2 + Buu + Bww + F11 + B22 (2.23)

where B2 = AB1 B1.IfFT2 C

TV1CF2 0, the Goh transformation is applied only on the singular partwhich is discussed in three cases. First, if some column vectors of CF2 is zero, by

rearranging the order of the column vectors of F2,

FT2 CTV1CF2 =

Q 00 0

(2.24)

where Q is positive definite. Then, 1 can be divided into 11 and 12 such that

(2.21) is singular with respect to 12, but not 11. The associated fault directions of

11 and 12 are denoted by B11 and B12, respectively. Then the Goh transformation

is applied only on 12,

22() =

t0

12(s)ds

2 = 1 B1222 (2.25)

Then,

2 = A2 + Buu + Bww + F11 + B22 (2.26)

where 2 = [ T11

T22 ]

T and B2 = [ B11 AB12 B12 ].The second case is that CF2 = 0, rank(CF2) < dim(CF2), and rank F2 = dim F2

which imply some column vectors of CF2 are the linear combinations of others.

Then, a new basis will be chosen for F2 such that some column vectors of the

new CF2 are zero and (2.24) is satisfied. The third case is that CF2 = 0 andrank F2 < dim F2 which imply some column vectors of F2 are the linear combina-

tions of others. Then, a new set of lower-order vectors can be formed for F2 such

that the new FT2 CTV1CF2 > 0. Note that it is possible that these three cases could

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happen at the same time. Nevertheless, there always exists a basis for F2 such that

either (2.24) is satisfied or FT2 CTV1CF2 > 0. By substituting (2.22) or (2.25) into

(2.20),

J = 12

t

t0

1 2Q1

1

w 2Q1w

2 2BT1CTV1CB1

y C2 2V1+(y C2)TV1CB12 + T2 BT1 CTV1(y C2)

d

12

2(t+0 ) + [ F2 B1 ][ 1(t+0 )T 2(t+0 )T ]T x0 20

Then, the new min-max problem is

min1

max2

maxw

max2(t

+0)

J

subject to (2.23) or (2.26).

The transformation process stops if the weighting on 2, BT1 C

TV1CB1, is posi-

tive definite. Otherwise, continue the transformation until there exists Bk such that

the weighting on k, BTk1C

TV1CBk1, is positive definite. Then, in the limit, the

min-max problem (2.4) becomes

min1

maxk

maxw

maxk(t+0 )

J

where

J =1

2

tt0

1 2Q1

1

w 2Q1w

k 2BTk1CTV1CBk1 y Ck 2V1

+(y Ck)TV1CBk1k + Tk BTk1CTV1(y Ck)

d

12 k(t+0 ) + B(t+0 ) x0 20 (2.27)

and B = [ F2 B1 B2 Bk1 ], = [ T1

T2

Tk ]

T

subject to

k = Ak + Buu + Bww + F11 + Bkk (2.28)

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Remark 5. The Goh transformation implies that there always exists a basis

for F2 such that the column vectors of F2 are output separable so that the out-

put separability test (1.14) remains valid. For time-invariant systems, the Goh

transformation is equivalent to the recursive algorithm (1.6) (Wonham, 1985).

The variational Hamiltonian of the problem is defined as

H = 12

1 2Q1

1

w 2Q1w

k 2BTk1CTV1CBk1 y Ck 2V1

+(y Ck)TV1CBk1k + Tk BTk1CTV1(y Ck)

+ T(Ak + Buu + Bww + F11 + Bkk)

where Rn is a continuously differentiable Lagrange multiplier. The first-ordernecessary conditions imply that the optimal strategies for 1, k, w and the dynamics

of are

1 = Q1FT1 (2.29a)

k = (BTk1C

TV1CBk1)1[BTk + B

Tk1C

TV1(y Ck)] (2.29b)

w = QwBTw (2.29c)

= AT CTV1(y Ck) + CTV1CBk1k (2.29d)

with boundary conditions,

(t+0 ) = (BT0B)1BT0[k(t+0 ) x0] (2.29e)

(t+0 ) = 0[k(t

+0 ) + B

(t+0 ) x0] (2.29f)

(t) = 0 (2.29g)

By substituting (2.29e) into (2.29f),

(t+0 ) = [0 0B(BT0B)1BT0][k(t+0 ) x0] (2.29h)

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By substituting (2.29a), (2.29b), (2.29c) into (2.28) and (2.29b) into (2.29d), a two-

point boundary value problem with boundary conditions (2.29g) and (2.29h) results

for satisfying

k

=

A Bk(BTk1C

T

V1

CBk1)1

BTk F1Q1F

T1 +BwQwB

Tw

CTHTV1HC AT

k

+

Buu + Bk(B

Tk1C

TV1CBk1)1BTk1C

TV1y

CTHTV1Hy

(2.30)

where A = A Bk(BTk1CTV1CBk1)1BTk1CTV1C and

H = I CBk1(BTk1CTV1CBk1)1BTk1CTV1 (2.31)

Note that k is now the state using the optimal strategies (2.29a), (2.29b) and (2.29c).

The form of (2.29h) suggests that

= S(k x) (2.32)

where

S(t+0 ) = 0 0B(BT0B)1BT0 (2.33a)

x(t+0 ) = x0 (2.33b)

and x is an intermediate state. By differentiating (2.32) and using (2.30),

0 =

S+ SA + ATS+ S

Bk(BTk1C

TV1CBk1)1BTk F1Q1FT1 + BwQwBTw

S

CTHTV1HCk

S+ ATS+ S[Bk(BTk1C

TV1CBk1)1BTk F1Q1FT1 + BwQwBTw ]S

x

Sx + SBuu + [SBk(BTk1CTV1CBk1)1BTk1CTV1 + CTHTV1H]y

By adding and subtracting SAx and CTHTV1HCx,

0 =

S+ SA + ATS+ S

Bk(BTk1C

TV1CBk1)1BTk F1Q1FT1 + BwQwBTw

S

CTHTV1HC (k x) Sx + SAx + SBuu+ [SBk(B

Tk1C

TV1CBk1)1BTk1C

TV1 + CTHTV1H](y Cx)

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Therefore, (2.32) is a solution to (2.30) if

Sx = SAx + SBuu + [SBk(BTk1C

TV1CBk1)1BTk1C

TV1

+ CTHTV1H](y

Cx) (2.34)

S = S[A Bk(BTk1CTV1CBk1)1BTk1CTV1C]

+ [A Bk(BTk1CTV1CBk1)1BTk1CTV1C]TS

+ S[Bk(BTk1C

TV1CBk1)1BTk F1Q1FT1 + BwQwBTw ]S

CTHTV1HC (2.35)

subject to (2.33).

By substituting 1

(2.29a), k

(2.29b), w (2.29c), (t+0

) (2.29e) and (2.32) into

the limiting cost criterion (2.27),

J =1

2

tt0

k x 2S[Bk(BTk1CTV1CBk1)1BTk F1Q1FT1 +BwQwBTw ]S

y Ck 2HTV1H

d 12

k(t+0 ) x0 200B(BT0B)1BT0

By adding the zero term

0 =1

2 k(t

+0 )

x0

2S(t+

0)

1

2 k(t)

x(t)

2S(t) +

1

2

t

t0

d

d k

x

2S d

to J,

J =1

2

tt0

k x 2S[Bk(BTk1CTV1CBk1)1BTk F1Q1FT1 +BwQwBTw ]S y Ck 2HTV1H +(Sk Sx)T(k x)

+(k x)TS(k x) + (k x)T(Sk Sx)

d

Note that k(t) x(t) 2S(t)= 0 because of the boundary condition (2.29g). By sub-stituting k (2.30), (2.32), (2.34), (2.35) into J

and expanding y Ck 2HTV1Hinto (y Cx) C(k x) 2HTV1H,

J = 12

tt0

y Cx 2HTV1H d

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Since k = x at current time t (2.29g), the limiting generalized least-squares fault

detection filter is (2.34). However, (2.34) can not be used because S has a null

space which is shown in Theorem 2.1. Therefore, a reduced-order filter for (2.34)

is derived in Section 2.6. Note that the second half of the optimization problem is

solved differently as (Chung and Speyer, 1998).

Theorem 2.1 shows that the limiting Riccati matrix S has a null space and satisfies

the sufficient condition (2.16) derived from the linear matrix inequality which implies

that S is the limit of .

Theorem 2.1.

S

Bk1 B1 F2 = 0S+ SA + ATS+ S(F1Q1FT1 + BwQwBTw)S CTV1C 0

Proof. By multiplying (2.35) by Bk1 from the right and subtracting SBk1 from

both sides,

d

d(SBk1) =

AT + S(F1Q1FT1 + BwQwBTw) + (SBk CTV1CBk1)

(BTk1CTV1CBk1)

1BTk SBk1This is a homogeneous differential equation and the boundary condition is zero be-

cause S(t+0 )B = 0 from (2.33a) and Bk1 is contained in B. Therefore SBk1 = 0.

Similarly, by multiplying (2.35) by Bk2 B1 and F2,

S

Bk2 B1 F2

= 0

To prove the second part of this theorem, (2.35) can be rewritten as

S+ SA + ATS+ S(F1Q1FT1 + BwQwBTw)S CTV1C

= (BTk S BTk1CTV1C)T(BTk1CTV1CBk1)1(BTk S BTk1CTV1C)

and it is nonpositive definite.

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2.5 Properties of the Null Space of S

In this section, some properties of the null space of S are given. It is shown that

the null space of S is equivalent to the minimal (C, A)-unobservability subspace for

time-invariant systems and a similar invariant subspace for time-varying systems.

Therefore, the limiting generalized least-squares fault detection filter is equivalent to

the unknown input observer and extends the unknown input observer to the time-

varying case. The minimal (C, A)-unobservability subspace ofF2 is the unobservable

subspace of (HC,A LC) (Massoumnia et al., 1989) for some filter gains L and

H:

Y Y, Ker H= CBk1 , H = I

CBk1[(CBk1)

TCBk1]1(CBk1)

T (2.36)

Note that Ker H = Ker H (2.31).

Theorem 2.2 shows that the null space of S is a (C, A)-invariant subspace. The-

orem 2.3 shows that the null space of S is contained in the unobservable subspace of

(HC,A LC).

Theorem 2.2. Ker S is a (C, A)-invariant subspace.

Proof. The dynamic equation of the error, e = x x, in the absence of the targetfault, process noise and sensor noise can be obtained by using (2.3) and (2.34).

Se = [SA SBk(BTk1CTV1CBk1)1BTk1CTV1C CTHTV1HC]e

because SF2 = 0. By adding Se to both sides and using (2.35),

d

d

(Se) =

[A

Bk(B

Tk1C

TV1CBk1)1BTk1C

TV1C]T

+S[Bk(BTk1C

TV1CBk1)1BTk F1Q1FT1 + BwQwBTw ]

Se (2.37)

If the error initially lies in Ker S, (2.37) implies that the error will never leave Ker S.

Therefore, Ker S is a (C, A)-invariant subspace.

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Theorem 2.3. Ker S is contained in the unobservable subspace of (HC,A LC).

Proof. Let Ker S. By multiplying (2.35) by T from the left and from theright,

d

d(TS) = TCTHTV1HC = 0

Then, HC = 0 because HC = 0 and Ker H = Ker H. From Theorem 2.2, Ker S is a

(C, A)-invariant subspace. Therefore, Ker S is contained in the unobservable subspace

of (HC,A LC).

From Theorem 2.1, CKer S CBk1. From Theorem 2.3, CKer S CBk1.Therefore, CKer S = CBk1 and H (2.7) is equivalent to H (2.36). Note that (2.36)

is a better way to form H which is used by the residual (2.6) because it does not

require the solution to the limiting Riccati equation (2.35).

For time-invariant systems, it is important to discuss the invariant zero directions

when designing the unknown input observer. The invariant zeros of (C,A,F2) will

become part of the eigenvalues of the filter if their associated invariant zero directions

are not included in the invariant subspace ofF2 (Massoumnia et al., 1989). Therefore,the null space of S needs to include at least the invariant zero directions associated

with the invariant zeros on the right-half plane and j-axis. However, the invariant

zeros on the left-half plane might become part of the filter eigenvalues since there is

no guarantee that their associated invariant zero directions are in the null space of

S. It is important that the left-half-plane invariant zeros are not part of the filter

eigenvalues because they might be ill-conditioned even though stable. This can be

done by modifying the nuisance fault directions to enforce the null space of S to

include the invariant zero directions. The invariant zero of (C,A,F2) is z at whichzI A F2

C 0

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Remark 6. The modification of the nuisance fault directions can apply to the

game-theoretic fault detection filter (Chung and Speyer, 1998) so that it could become

equivalent to the unknown input observer in the limit.

Remark 7. In order to detect the target fault, F1 can not intersect the null space of

S which is unobservable to the residual. If it does, the target fault will be difficult or

impossible to detect even though the filter can still be derived by solving the min-max

problem.

2.6 Reduced-Order Filter

In this section, reduced-order filters are derived for the limiting generalized least-

squares fault detection filter (2.34) for both time-varying and time-invariant systems.

The reduced-order filter is necessary for implementation because (2.34) can not be

used due to the null space of S. It is shown that the reduced-order filter completely

blocks the nuisance fault.

Since S(t) is non-negative definite, there exists a state transformation (t) such

that

(t)TS(t)(t) =

S(t) 0

0 0

(2.39)

where S(t) is positive definite. Theorem 2.4 provides a way to form the transformation

(t).

Theorem 2.4. There exists a state transformation (t) where

Z(t) Ker S(t)

= (t)

Z1(t) 0

0 Z2(t)

(2.40)

Z is any n (n k2) continuously differentiable matrix such that itself and Ker Sspan the state space where n = dim X and k2 = dim(Ker S). Z1 and Z2 are any

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(n k2) (n k2) and k2 k2 invertible continuously differentiable matrices, re-spectively. Then, the (t) obtained from (2.40) satisfies (2.39).

Proof. From (2.40),

Ker S =

0

Z2

S

0

Z2

= 0 TS

0

Z2

= 0

Since Z2 is invertible by definition and TS is symmetric, (2.39) is true.

Note that Theorem 2.4 does not define uniquely and can be computed apriori

because Ker S can be obtained apriori.

By applying the transformation to the estimator states,

1x= =

12

By multiplying (2.34) by T from the left, adding TS1x to both sides, and using

1 = I,

S 00 0

12

=

S 00 0

1

12

+

S 00 0

A11 A12A21 A22

12

+

S 00 0

M1M2

u

+

S 00 0

G1G2

DT1 DT2

C

T

1CT2

V1

C1 C2

D1D2

1DT1 DT2

C

T

1CT2

V1

+

CT1CT2

HTV1H

y C1 C2

12

(2.41)

where

1A =

A11 A12A21 A22

, 1Bu =

M1M2

, C =

C1 C2

1Bk1 = D1

D2 , 1Bk = G1

G2

Since SBk1 = 0 from Theorem 2.1,

TS1Bk1 =

SD1

0

= 0

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which implies D1 = 0. Then, (2.41) can be transformed into two equations,

S1 = S(A1111)1 + S(A1212)2 + SM1u+[SG1(DT2 CT2 V1C2D2)1DT2 CT2 V1

+CT1 HTV1H](y

C11

C22) (2.42a)

0 = CT2 HTV1H(y C11 C22) (2.42b)

where

1 =

11 1221 22

Note that 1 and can be computed apriori from (2.40).

By multiplying (2.35) by T from the left and from the right, subtracting TS

and ST to both sides, and using 1 = I, the limiting Riccati equation can be

transformed into two equations,

0 = S[A12 12 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C2] (2.43)

S = S[A11 11 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C1]

+ [A11 11 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C1]TS

+S

[G

1(DT

2CT2

V1C2

D2)

1GT1

N1

Q1

NT1 +

R1

Qw

RT1 ]

S

CT1 HTV1HC1 (2.44)

where

1Bw =

R1R2

, 1F1 =

N1N2

and

(t+0 )TS(t+0 )(t+0 ) =

S(t+

0 ) 00 0

From (2.42b),

HC2 = 0 (2.45)

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where e1 = 1 1. From (2.43),

A12 12 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C2 = 0 (2.49)

because S is positive definite. By using (2.46), (2.48) and (2.49),

e1 = [A11 11 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C1 S1CT1 HTV1HC1]e1+ R1w + N11 [G1(DT2 CT2 V1C2D2)1DT2 CT2 V1 + S1CT1 HTV1H]v

This shows that the residual is not affected by the nuisance fault.

For time-invariant systems, the reduced-order limiting filter and reduced-order

limiting Riccati equation can be derived similarly.

1 = A111 + M1u + [G1(DT2 C

T2 V

1C2D2)1DT2 C

T2 V

1

+ S1CT1 HTV1H](y C11) (2.50)

S = S[A11 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C1]

+ [A11 G1(DT2 CT2 V1C2D2)1DT2 CT2 V1C1]TS

+ S[G1(DT2 C

T2 V

1C2D2)1GT1 N1Q1NT1 + R1QwRT1 ]S

CT1 HTV1HC1 (2.51)

For time-invariant systems, (2.40) is constant because Ker S is fixed. Ker S can

be obtained from computing the minimal (C, A)-unobservability subspace ofF2 (1.8)

instead of solving (2.35).

2.7 Example

In this section, three numerical examples are used to demonstrate the performance

of the generalized least-squares fault detection filter. In Section 2.7.1, the filter is

applied to a time-invariant system. In Section 2.7.2, the filter is applied to a time-

varying system. In Section 2.7.3, the null space of the limiting Riccati matrix S

(2.35) is discussed.

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2.7.1 Example 1

In this section, two cases for a time-invariant problem are presented. The first one

shows that the sensitivity of the filter (2.12) to the nuisance fault decreases when is

smaller. The second one shows that the sensitivity of the reduced-order limiting filter

(2.50) to the target fault increases when Q1 is larger. The time-invariant system is

from (White and Speyer, 1987).

A =

0 3 41 2 3

0 2 5

, C = 0 1 0

0 0 1

, F1 =

00

1

, F2 =

51

1

where F1 is the target fault direction and F2 is the nuisance fault direction. There isno process noise.

In the first case, the steady-state solutions to the Riccati equation (2.13) are

obtained with weightings chosen as Q1 = 1, Q2 = 1, and V = I when = 104 and

106, respectively. Figure 2.1 shows the frequency response from both faults to the

residual (2.6). The left one is = 104, and the right one is = 106. The solid

lines represent the target fault, and the dashed lines represent the nuisance fault.

This example shows that the nuisance fault transmission can be reduced by using a

smaller while the target fault transmission is not affected.

In the second case, the steady-state solutions to the reduced-order limiting Riccati

equation (2.51) are obtained with V = 104I when Q1 = 0 and 0.0019, respectively.

Figure 2.2 shows the frequency response from the target fault and sensor noise to the

residual (2.47). The left one is Q1 = 0, and the right one is Q1 = 0.0019. The solid

lines represent the target fault, and the dashed lines represent the sensor noise. Thisexample shows that the sensitivity of the filter to the target fault can be enhanced

by using a larger Q1. The sensor noise transmission also increases because part of

the sensor noise comes through the same direction as the target fault. However,

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10-2

100

102

-180

-160

-140

-120

-100

-80

-60

-40

-20

0gamma = 10^(-6)

Frequency (rad/s)

Singularvalue(db)

10-2

100

102

-180

-160

-140

-120

-100

-80

-60

-40

-20

0gamma = 10^(-4)

Frequency (rad/s)

Singularvalue(db)

Figure 2.1: Frequency response from both faults to the residual

the sensor noise transmission is small compared to the target fault transmission. In

either case, the nuisance fault transmission stays zero and is not shown in Figure 2.2.

Note that when Q1 = 0, the generalized least-squares fault detection filter is similar

to (Chung and Speyer, 1998) which does not enhance the target fault transmission.

2.7.2 Example 2

In this section, the filter (2.12) and the reduced-order limiting filter (2.46) are applied

to a time-varying system which is from modifying the time-invariant system in the

previous section by adding some time-varying elements to A and F2 matrices while

C and F1 matrices are the same.

A = cos(t) 3 + 2sin(t) 41 2 3 2cos(t)

5sin(t) 2 5 + 3cos(t)

, F2 = 5 2cos(t)11 + sin(t)

The Riccati equation (2.13) is solved with Q1 = 1, Q2 = 1, V = I and = 10

5 for

t [0, 25]. The reduced-order limiting Riccati equation (2.44) is solved with the same

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10-2

100

102

-70

-60

-50

-40

-30

-20

-10

0

10

20Q1 = 0

Frequency (rad/s)

Singularvalue(db)

10-2

100

102

-70

-60

-50

-40

-30

-20

-10

0

10

20Q1 = 0.0019

Frequency (rad/s)

Singularvalue(db)

Figure 2.2: Frequency response from the target fault and sensor noise to the residual

Q1 and V. Figure 2.3 shows the time response of the norm of the residual when there

is no fault, a target fault and a nuisance fault, respectively. The faults are unit steps

that occur at the fifth second. In each case, there is no sensor noise. The left three

figures show the residual (2.6) for the filter (2.12). There is a small nuisance fault

transmission because (2.12) is an approximate unknown input observer. The right

three figures show the residual (2.47) for the reduced-order limiting filter (2.46). Note

that the nuisance fault transmission is zero. There is a transient response until about

two seconds due to the initial condition. This example shows that both filters, (2.12)

and (2.46), work well for time-varying systems.

2.7.3 Example 3

In this section, three cases are presented to show the properties of the null space of

the limiting Riccati matrix S. The first case shows that Ker S includes the nuisance

fault direction and the invariant zero direction associated with the right-half-plane

invariant zero. The second case shows that Ker S includes only the nuisance fault

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0 5 10 15 20 250

0.5

1x 10

-3 No fault

Residual

0 5 10 15 20 250

0.2

0.4

0.6Target fault

Residual

0 5 10 15 20 250

0.5

1x 10

-3 Nuisance fault

Time (sec)

Residual

In the limit

0 5 10 15 20 250

0.5

1x 10

-3 No fault

Residual

0 5 10 15 20 250

0.2

0.4

0.6Target fault

Residual

0 5 10 15 20 250

0.5

1x 10

-3 Nuisance fault

Time (sec)

Residual

Not in the limit

Figure 2.3: Time response of the residual

direction, but not the invariant zero direction associated with the left-half-plane in-

variant zero. The third case shows that the invariant zero direction associated with

the left-half-plane invariant zero is included in Ker S if the nuisance fault direction

is modified. These three cases show that the null space of S is equivalent to the

minimal (C, A)-unobservability subspace of F2.

In the first case, A and C matrices are the same as the example in Section 2.7.1

and

F1 =

10.5

0.5

, F2 =

31

0

(C,A,F2) has an invariant zero at 3 and the invariant zero dire

Documents

Chen Dissertation